# ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals

arXiv: https://arxiv.org/abs/2503.08305 · HTML: https://mlsanity.com/paper/2503.08305/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 90 references that are in the map.
- First posted on arXiv 11 Mar 2025.

## Builds on (most cited first, 30 of 35)

- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Object-Centric Learning with Slot Attention](https://mlsanity.com/paper/2006.15055/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Graph Normalizing Flows](https://mlsanity.com/paper/1905.13177/) (2019)
- [Kernel Operations on the GPU, with Autodiff, without Memory Overflows](https://mlsanity.com/paper/2004.11127/) (2020)
- [SpookyNet: Learning Force Fields with Electronic Degrees of Freedom and Nonlocal Effects](https://mlsanity.com/paper/2105.00304/) (2021)
- [Generating 3D Molecules Conditional on Receptor Binding Sites with Deep Generative Models](https://mlsanity.com/paper/2110.15200/) (2021)
- [FAENet: Frame Averaging Equivariant GNN for Materials Modeling](https://mlsanity.com/paper/2305.05577/) (2023)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials](https://mlsanity.com/paper/2306.06482/) (2023)
- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [Relaxing Equivariance Constraints with Non-stationary Continuous Filters](https://mlsanity.com/paper/2204.07178/) (2022)
- [Ab-Initio Potential Energy Surfaces by Pairing GNNs with Neural Wave Functions](https://mlsanity.com/paper/2110.05064/) (2021)
- [Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks](https://mlsanity.com/paper/2007.02005/) (2020)
- [Informing Geometric Deep Learning with Electronic Interactions to Accelerate Quantum Chemistry](https://mlsanity.com/paper/2105.14655/) (2021)
- [Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids](https://mlsanity.com/paper/2112.00652/) (2021)
- [Symmetry Breaking and Equivariant Neural Networks](https://mlsanity.com/paper/2312.09016/) (2023)
- [Laplacian Canonization: A Minimalist Approach to Sign and Basis Invariant Spectral Embedding](https://mlsanity.com/paper/2310.18716/) (2023)
- [Approximately Equivariant Graph Networks](https://mlsanity.com/paper/2308.10436/) (2023)
- [Crystalformer: Infinitely Connected Attention for Periodic Structure Encoding](https://mlsanity.com/paper/2403.11686/) (2024)
- [Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materials](https://mlsanity.com/paper/2312.05388/) (2023)
- [DeepDFT: Neural Message Passing Network for Accurate Charge Density Prediction](https://mlsanity.com/paper/2011.03346/) (2020)
- [Equivariant Neural Operator Learning with Graphon Convolution](https://mlsanity.com/paper/2311.10908/) (2023)

## Built on by (most cited first, 10 of 10)

- [Towards A Transferable Acceleration Method for Density Functional Theory](https://mlsanity.com/paper/2509.25724/) (2025)
- [Global Plane Waves From Local Gaussians: Periodic Charge Densities in a Blink](https://mlsanity.com/paper/2601.19966/) (2026)
- [CORDS: Continuous Representations of Discrete Structures](https://mlsanity.com/paper/2601.21583/) (2026)
- [M=oLe-Lambda: Learning the Coupled-Cluster Response State for Energies, Gradients, and Properties](https://mlsanity.com/paper/2605.29622/) (2026)
- [ED-DiT: Physics-Guided Diffusion Pretraining for Transferable Molecular Representations from Electron Density](https://mlsanity.com/paper/2608.03260/) (2026)
- [Coupled Cluster con M=oLe: Molecular Orbital Learning for Neural Wavefunctions](https://mlsanity.com/paper/2602.20232/) (2026)
- [Neural-Network Solutions to Real-Space Charge Density and Generalization](https://mlsanity.com/paper/2609.14906/) (2026)
- [Complete Neural Electronic Initialization Accelerates Materials DFT](https://mlsanity.com/paper/2609.21759/) (2026)
- [Scaling Density Functional Theory with Gaussian Splatting](https://mlsanity.com/paper/2609.31483/) (2026)
- [Equivariant Flow Matching for Electron Density Prediction](https://mlsanity.com/paper/2610.02651/) (2026)

## Abstract (arXiv)

We present the Electronic Tensor Reconstruction Algorithm (ELECTRA) - an equivariant model for predicting electronic charge densities using floating orbitals. Floating orbitals are a long-standing concept in the quantum chemistry community that promises more compact and accurate representations by placing orbitals freely in space, as opposed to centering all orbitals at the position of atoms. Finding the ideal placement of these orbitals requires extensive domain knowledge, though, which thus far has prevented widespread adoption. We solve this in a data-driven manner by training a Cartesian tensor network to predict the orbital positions along with orbital coefficients. This is made possible through a symmetry-breaking mechanism that is used to learn position displacements with lower symmetry than the input molecule while preserving the rotation equivariance of the charge density itself. Inspired by recent successes of Gaussian Splatting in representing densities in space, we are using Gaussian orbitals and predicting their weights and covariance matrices. Our method achieves a state-of-the-art balance between computational efficiency and predictive accuracy on established benchmarks. Furthermore, ELECTRA is able to lower the compute time required to arrive at converged DFT solutions - initializing calculations using our predicted densities yields an average 50.72 % reduction in self-consistent field (SCF) iterations on unseen molecules.